在这里单连通的假设是有用的。
And, that's where the assumption of simply connected is useful.
全空间去掉原点，它是单连通的吗？
OK, so space with the origin removed, OK, you think it's simply connected?
这就是球面上，对单连通的定义。
OK, so that's the definition of the surface of a sphere being simply connected.
总之，这是关于单连通的习惯性提醒。
OK, so anyway, that's the customary warning about simply connected things.
谁认为它不是单连通的？
这是检验,该区域不是单连通的一种方法。
So, that's one way to check just for sure that this thing is not simply connected.
谁认为它是单连通的？
好滴，目前这里，在有定义的区域里，减掉原点，即移去原点，就不是单连通的。
OK, so for this guy, domain of definition, which is plane minus the origin with the origin removed is not simply connected.
在这个例子里，圈饼表面就是非单连通的，并且这类物体的简化就会像是至少有一个洞的曲面。
In this example, the surface is not simply connected and any smoothed-out object looks like a torus with at least one hole.
他在1904年就提出猜想：在四维空间，所有单连通的封闭三维面都能转化为一个三维球体。
His conjecture, made in 1904, was that in this four-dimensional world, all closed three-dimensional surfaces that are simply connected could be transformed to look like a three-dimensional sphere.
单连通和非单连通的概念，研究哪些环路能界定曲面，可以用来对空间内部物体的形状进行分类。
This concept of being simply connected or not, and studying which loops bound surfaces or not can be used to classify shapes of things inside space.
讨论了紧致非单连通的具非负曲率的流形的一些几何性质，并应用它们证明了具非负曲率的紧致非单连通曲面必为平坦的。
With the help of them, it can be proved that the non-simply connected compact surface with nonnegative curvature must be flat.
因为我已经告诉过你们空间是单连通的，所以，如果我们真的想要运用Stokes定理，那么我总是可以把它运用于任何曲线，我应该做什么呢？
So, if we really wanted to apply Stokes theorem, because I've been telling you that space is simply connected, and I will always be able to apply Stokes theorem to any curve, what would I do?
那么我们是在哪里利用了，“定义在单连通区域”的假设呢？
OK, so where do we use the assumption of being defined in a simply connected region?
这里，为了使得完全成立，不得不假设，这定义在一个单连通区域，否则就有可能得到，和先前一样的奇怪事情。
And here, to be completely truthful, I have to say defined in a simply connected region. Otherwise, we might have the same kind of strange things happening as before.
期末考也可能会用得上，实际上，真的不会有十分复杂的问题，不过你们也要大概地了解一下,这个关于单连通区域的知识点。
And, maybe on the final it would be, there won't be any really, really complicated things probably, but you might need to be at least vaguely aware of this issue of things being simply connected.
一个向量场,如果定义在单连通区域并且旋度为零，那么它就是一个梯度场，并且其上的线积分与路径无关。
OK, so, we've seen that if we have a vector field defined in a simply connected region, and its curl is zero, then it's a gradient field, and the line integral is path independent.
该算法能处理多个区域的情况和区域为单连通或复连通区域的情况；
The improved algorithm can deal with multiple regions, single connected domain and multiple connected domain.
并由此讨论了紧致的非单连通黎曼流形上无穷多的闭测地线存在性问题。
Also, the paper discuss the existence of the infinite closed geodesics of a compact no-simply connected Riemannian manifold.
对比传统单连通区域扫描线填充法，新方法算法效率高，实现简单，对复杂区域的填充同样适用。
Compared with the normal algorithms, the new one is efficient in algorithm, simple in realization for scan filling of simple connected area and also applicable to the filling of the complex area.
将单值映射的弧连通凸概念推广到了集值映射。
This paper extends the concept of arc connected convexness of single valued maps to set valued maps.
所提取的骨架彼此连通、单像素宽并与原始图像拓扑一致。
It is connected and single-pixel wide, and keeps the topological properties of the original object image.
本文把在单连通区域上成立的曲线积分与路线无关性定理推广到复连通区域。
In this paper the theorem in which a curve integral is independent of the integral path on a single connected region is generalized.
该算法能统一处理互感和无互感线路、单连通树支集的特定网络以及树枝集中有分支的通用电网络。
The algorithm can deal with mutual inductance or non mutual inductance network and especial 'a stroke' tree set network or general offset-trees network.
证明了皱褶立方体的单连通性与一致局部单连通性。
We prove that the crumpled cube is simply-connected and uniformly locally simply-connected.
提出一种基于边界标注的单连通区域扫描线填充快速算法。
A novel edge - labeled algorithm for simple connected area scan filling is presented.
给出了单连通区域和多连通区域中的一般级数表达式及确定级数项系数的方法。
A series expansion and a method to determine the coefficients of the series expansion are given.
本文提出一种算法实现单连通区域的线性四元树表示转换成区域边界的4 -方向链码描述。
An algorithm is presented for converting the linear quadtree representation of a simply connected region into a 4-direction chain code description of the region's boundary.
本文是作者在多连通区域单叶函数领域研究成果的总结。
The present article is an account of results on univalent functions in multiply connected domains obtained by the author.
把单值边缘连续映射的概念推广到集值映射上，并且给出了集值边缘连续映射是连续映射与连通映射的条件。
The concept of peripherally continuous mappings is extended to set-value and the conditions are given which set-value peripherally continuous mappings are continuous and connected mappings.
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